Can the government force private schools to use Filipino as medium of instruction?

To a large extent, yes. The private school may claim the right to use the regional language as auxiliary medium of instruction, but as what is meant by "auxiliary medium of instruction" and "regional language" may be thrown out by the Supreme Court as political and nonjudicial questions (and therefore it cannot decide on them).

However, since most private schools which are in this position are using English for their instruction, it would be a very difficult fight. I wonder how the court could be persuaded that English is one's regional language.