Making the plane infinite makes the problem easier, since you don’t need to worry about the boundary of any finite section of the plane. For example, if you choose a triangular section, you won’t be able to tile that section by squares. But it is still clear the infinite plane can be tiled by squares; just keep laying the squares edge-to-edge.

You can cover an infinite plane easily enough. You just need an infinite number of tiles. And that's no problem in math. It's no different from saying, for example, that the intervals of form [n,n+1), where n is an integer, cover the whole real number line.

Afaik the tiling should be able to fill a finite plane of arbitrary size. Thus, the same tiling should also be able to go on forever.

The way many things relating to infinity in math are described is something like this:

You get to pick a size of tile. You can pick whatever you want, totally independently. 700x700, 1000000x1000000, whatever, it's fine.

If I can describe a mapping for that arbitrary size tile, then I'm able to map an "infinitely" large plane. That even as we approach infinity, no matter the size of the finite plane, I can make an arrangement of tiles that fully covers the plane, means I've "infinitely" mapped the plane. (In some cases where there's uncertainty you might say something more like we've shown that the limit as plane size approaches infinity is mappable.)

"Does the plane have to be a plane in the mathematical sense, i.e. infinite? Because then you're obviously not going to cover it with anything. It goes on forever."

You certainly can tile an infinite plane. In fact, you can view this algebraically if it helps, by talking about tiling the complex plane. For example, the Gaussian Integers, which are complex numbers of the form (a+bi) where a and b are integers, can be viewed as the corners of squares that tile the complex plane.

Border is irrelevant, for what it's worth it can be whatever you want