In _Keyishian v. Board of Regents_ (1967), the Court addressed the
question of what kind of loyalty oaths government employees (specially
SUNY faculty members) could be required to sign as a condition of
employment. The Court said: "Thus [the New York laws] suffer from
impermissible 'overbreadth'. They seek to bar employment both for
association which legitimately may be sanctioned and for association
which may not be sanctioned consistently with First Amendment rights.
[The laws] are [therefore] invalid insofar as they sanction mere
knowing membership without any showing of specific intent to further
the unlawful aims of [the Communist Party]."

[This is the case referenced in the university-free-expression-is-
fundamental-to-society part of _Rust v.  Sullivan_.]