Magnetic flux is quantized if it is produced by a superconducting
persistent current. By the same reasoning that leads to the quantization
of angular momentum (or to the single-valuedness of wave functions in
the azimuthal direction that David Bowman mentioned earlier) electron
pairs (Cooper pairs) circulating in a superconducting ring can produce
only discrete values of magnetic flux by Ampere's law. The flux quantum
observed (in 1961 by Deaver and Fairbank at Stanford and also by Doll
and Nabauer) demonstrated that the charge carrying entity in
superconductors has a charge of 2e rather than e, supporting Cooper's
mechanism which led to the BCS theory of superconductivity.
In other circumstances magnetic flux need not be quantized. By Ampere's
law it can be as continuously variable as the current which produces it.