How did "squarely" come to mean "in a straight or straightforward manner"?
Squarely is the adv. of Square. Its meaning is in a square shape, form, which means the shape has 90 degree.
However, it has other meanings: directly; without evasion; in a straight or straightforward manner
So it is extremely weird: How does it come form "square shape with 90 degree" to "directly"?
Square is originally a tool measuring right angles from a1300 (according to OED's earliest citation). The tool helps you to construct something solidly and properly. Thus, the adjective square and adverb squarely are associated with being fair, honest, honourable and straightforward.
This is briefly mentioned in the website saywhydoi.com also:
Why would a square be associated with honesty and doing things properly?
It may be connected to the fact that a square is not only the geometric shape we’re familiar with. Since the 13th century it’s also been the term describing the tool used to measure right angles. Right angled shapes like squares and rectangles may have been considered as “right” partly because they helped stonemasons ensure they created solid, steady, stable constructions.
I'm quite sure it comes from the idea of facing someone very directly, face-to-face - as opposed to obliquely, which is exactly what a square isn't.
If you picture the geometry of two people facing each other directly, there are plenty of right angles there.
A square is a shape without 'complication' (something that folds, wraps, or twists together; something that intertwines or entangles one with another or adds difficulties; a complicating factor OED).
There is nothing equivocal or ambiguous about a square. It's nature is precise and easily understood. It is made by putting together four lines of equal length, each at the same right-angle to its neighbour.
For these reasons squarely is used to describe activities that are conducted honestly, fairly, and in a straightforward manner.
"Be there or be square, man" (https://en.wikipedia.org/wiki/Square_(slang)). In '60's counter culture speak 'square' is so straight-forward, safe, predictable and dull that it's pejorative.