I'm wondering why you don't think nature should be allowed to take its course. Eagles are not at all threatened any more, and beavers provide great habitat for all sorts of species. I also don't think that beavers normally chop down trees large enough to support an eagles nest (could be wrong on this).
If you insist on trying something, I'd suggest talking to MNR to see what leeway you have. Is it on your own land? Otherwise I would be extremely surprised if anything could be done unless roads or buildings were threatened.