It's possible to have an equiangular quadrilateral, i.e. whose sides are geodesics (the analogue of "straight line" on a sphere). The Gauss-Bonnet theorem implies their total interior angle is greater than 2pi, so four right angles can't work.
Here's an interactive demo of quadrilaterals on the sphere: https://geogebra.org/m/q83rUj8r
Notice that each side is a segment of a great circle, i.e. a circle that divides the sphere in half. That's what it means for a path to be a geodesic on the sphere.

One of the few things I can be pedantic about, so I must...
Ambient isotopic in R^3, which is much stronger than homeomorphic, which is the usual notion of topological equivalence. Yet easier to understand intuitively, because ambient isotopy classes are basically just "what you can do with a rubber band."