Using "-Filter" with a variable

You don't need quotes around the variable, so simply change this:

Get-ADComputer -Filter {name -like '$nameregex' -and Enabled -eq "true"}

into this:

Get-ADComputer -Filter {name -like $nameregex -and Enabled -eq "true"}

Note, however, that the scriptblock notation for filter statements is misleading, because the statement is actually a string, so it's better to write it as such:

Get-ADComputer -Filter "name -like '$nameregex' -and Enabled -eq 'true'"

Related. Also related.

And FTR: you're using wildcard matching here (operator -like), not regular expressions (operator -match).