Play with squares and circles

I can fit at least 156 squares in the circle, as follows.

Row #1: place 14 squares side by side, with their top aligned with the horizontal diameter of the circle. This fits, because the width of the circle 20 mm below the diameter is still $2\cdot \sqrt{150^2 - 20^2} \approx 297.3 > 14\cdot 20$ .

Row #2: place another 14 squares side by side with their top aligned with the bottom of row 1. This still fits, because the width of the circle 40 mm below the diameter is $2\cdot \sqrt{150^2 - 40^2} \approx 289.14 > 14\cdot 20$ .

Row #3: 13 squares next to each other, directly underneath row #2: $2\cdot \sqrt{150^2 - 60^2} \approx 274.9 > 13\cdot 20$

And so on: row #4 has 12 squares; row #5 has 11 squares; row #6 has 9 squares (the first and last touch the circle exactly at the far bottom corners); finally, row #7 has 5 squares. This still fits because $2\cdot \sqrt{150^2 - 140^2} \approx 107.7 > 5\cdot 20$ .

This adds up to 78 squares filling the bottom half of the circle. Cover the top half in the same way.

Note that I have not proven that 156 is the correct maximum; only that the maximum is at least 156. (I did check a method that starts with a row symmetrically around the center diameter, and that fits only 154 squares.)