I realize that the sphere wouldn't be able to see the square because he is infinitely thin, but wouldn't the square be able to see the sphere if he passed into his plane?
No. Between any pair of atoms there is a gap containing no matter, and this gap is on the order of 100,000 times bigger than an individual atom. Because of this, it's pretty much impossible for Sphere's atoms to happen to line up in just the right way for them to all intersect with Square's plane. Even if they did, Sphere would have to be at a temperature of absolute zero, otherwise vibration from heat would make them not line up again.
And what I was really trying to say is this: could any hypothetical being of n-1 dimensions fully imagine n dimensions, or is it truly impossible?
Yes. For example, you could have a dream which was in 4D. It's just highly unlikely, because our dreams are mainly built on what we know of real life.