I came to the United States on an H1B visa in 1987 and my employer did not withhold any FICA payments until mid-1990. My former employer was purchased by another company in the mid-90s and since then continues to exist but has been changed ownership and names a few times. I was unaware of this until recently and am wondering what my liability would be and how I could remedy this situation.

There is NO tax liability on either your part or the former employer's part.

This happened over 20 years ago, and the statute of limitation on this has passed. Since the IRS no longer has any legal standing to collect these taxes, you have NOTHING to worry about.