How can I have two fixed width columns with one flexible column in the center?

Instead of using width (which is a suggestion when using flexbox), you could use flex: 0 0 230px; which means:

• 0 = don't grow (shorthand for flex-grow)
• 0 = don't shrink (shorthand for flex-shrink)
• 230px = start at 230px (shorthand for flex-basis)

which means: always be 230px.

See fiddle, thanks @TylerH

Oh, and you don't need the justify-content and align-items here.

img {
    max-width: 100%;
}
#container {
    display: flex;
    x-justify-content: space-around;
    x-align-items: stretch;
    max-width: 1200px;
}
.column.left {
    width: 230px;
    flex: 0 0 230px;
}
.column.right {
    width: 230px;
    flex: 0 0 230px;
    border-left: 1px solid #eee;
}
.column.center {
    border-left: 1px solid #eee;
}

Despite setting up dimensions for the columns, they still seem to shrink as the window shrinks.

An initial setting of a flex container is flex-shrink: 1. That's why your columns are shrinking.

It doesn't matter what width you specify (it could be width: 10000px), with flex-shrink the specified width can be ignored and flex items are prevented from overflowing the container.

I'm trying to set up a flexbox with 3 columns where the left and right columns have a fixed width...

You will need to disable shrinking. Here are some options:

.left, .right {
     width: 230px;
     flex-shrink: 0;
 }  

OR

.left, .right {
     flex-basis: 230px;
     flex-shrink: 0;
}

OR, as recommended by the spec:

.left, .right {
    flex: 0 0 230px;    /* don't grow, don't shrink, stay fixed at 230px */
}

7.2. Components of Flexibility

Authors are encouraged to control flexibility using the flex shorthand rather than with its longhand properties directly, as the shorthand correctly resets any unspecified components to accommodate common uses.

More details here: What are the differences between flex-basis and width?

An additional thing I need to do is hide the right column based on user interaction, in which case the left column would still keep its fixed width, but the center column would fill the rest of the space.

Try this:

.center { flex: 1; }

This will allow the center column to consume available space, including the space of its siblings when they are removed.

Revised Fiddle