Nooooooo!!! Very bad idea. Whilst it is possible to come here and stay without the proper papers and do almost whatever the hell you want, general aviation training is very high on the TSA radar.
And with today's 'buzzing' of the White House by some idiot in a 152 (twice!) it is likely only to get worse.
Like everyone says - get the right visa, and avoid any school which says they'll take you without it.
Paul