Packing rectangles into convex polygons but without rotations
The problem can be reformulated as picking a maximum number of points inside a convex polygon, such that the every pair of them is in distance (under the $L_\infty$ metric) at least $1$ from each other (just think about the centers of the squares). This in turn is related to the same problem where one uses the regular Euclidean distance. This is in turn related to meshing, where one is interested in breaking a polygon into nicely behaved regions (i.e., you take the Voronoi diagram of the centers [see Centroidal Voronoi tessellations]).

Anyway, a $(1-\epsilon)$-approximation is quite easy. You randomly slide a grid of sidelength $O(1/\epsilon)$. Clip the polygon into the grid, and solve the problem inside each piece of intersection of the polygon with the grid using brute force. An algorithm with running time $O(M*noise(\epsilon))$ should easily follow, where $M$ is the number of points (i.e., rectangles), and $noise(\epsilon)$ is some horrendous function that depends only on $\epsilon$.