css grid of squares with flexbox

you should not set any size. you may use an extra element or a pseudo elemnt with vertical padding in %. this will allow you to use width as reference : a snippet to show:

.flex-container {
    padding: 0;
    margin: 0;
    list-style: none;
    display: -webkit-box;
    display: -moz-box;
    display: -ms-flexbox;
    display: -webkit-flex;
    display: flex;
    -webkit-flex-flow: row;
    justify-content: space-around;
   
    line-height:30px;
}
.flex-item {
    background: tomato;
    margin: 5px;
    color: white;
    font-weight: bold;
    font-size: 1.5em;
    text-align: center;
    flex: 1 0 auto;
    height:auto;
}
.flex-item:before {
    content:'';
    float:left;
    padding-top:100%;
}
<body>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
</body>

an inline-block element could do too, just adapt the display/behavior of the box and it's content. the magic here comes from padding:50% 0; (100% vertical padding equals width of parent). see w3c about vertical margin and padding

[edit 07/2021]about centering the content inside that square for who ever needs this too (make the square itself a flex boxe too):

.flex-container {
    padding: 0;
    margin: 0;
    list-style: none;
    display: -webkit-box;
    display: -moz-box;
    display: -ms-flexbox;
    display: -webkit-flex;
    display: flex;
    -webkit-flex-flow: row;
    justify-content: space-around;
   
    line-height:30px;
}
.flex-item {
    background: tomato;
    margin: 5px;
    color: white;
    font-weight: bold;
    font-size: 1.5em;
    text-align: center;
    flex: 1 0 auto;
    height:auto;
    
    display:flex;
    align-items:center;
    justify-content:center;
}
.flex-item:before {
    content:'';
    padding-top:100%;
}
<body>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
    <div class="flex-container">
        <div class="flex-item">1</div>
        <div class="flex-item">2</div>
        <div class="flex-item">3</div>
        <div class="flex-item">4</div>
        <div class="flex-item">5</div>
        <div class="flex-item">6</div>
        <div class="flex-item">7</div>
    </div>
</body>

You may also think interesting : Responsive grid of squares within a responsive grid of squares & 4x4 grid of squares that scale up to a maximum width if this answer do not fully suits your square needs ;)


For those that also want to use display: flex inside the square divs, you need to use display: table for the :before element, otherwise the square will work with Chrome but won't work with Firefox or Edge (as of Firefox 47 and Edge 13).

In the snippet bellow, which should work for all browsers, I also demonstrate how to wrap unlimited items with percentage columns (in this case 20%) and separate them with padding and inner divs, since margins with percentages do not work correctly in FF and of course Edge.

.flex-container {
  display: flex;
  justify-content: start;
  flex-wrap: wrap; 
}

.flex-cell {
  flex: 0 0 20%;
  display: flex;
  justify-content: center;
  align-items: stretch;
  padding: 0.5rem;
  box-sizing: border-box;
}

.flex-cell:before {
  content: '';
  display: table;
  padding-top: 100%;
}

.flex-item {
  flex-grow: 1;
  border: 1px solid black;
  background: tomato;
  color: white;
  
  display: flex;
  justify-content: center;
  align-items: center;
}
<body>
    <div class="flex-container">
        <div class="flex-cell">
            <div class="flex-item">1</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">2</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">3</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">4</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">5</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">6</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">7</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">8</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">9</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">10</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">11</div>
        </div>
        <div class="flex-cell">
            <div class="flex-item">12</div>
        </div>
    </div>
</body>