We present a sample of 34 normal Type II supernovae (SNe II) detected with the Zwicky Transient Facility, with multiband UV light curves starting at t ≤ 4 days after explosion, and X-ray observations. We characterize the early UV-optical color, provide empirical host-extinction corrections, and show that the t > 2 day UV-optical colors and the blackbody evolution of the sample are consistent with shock cooling (SC) regardless of the presence of “flash ionization” features. We present a framework for fitting SC models that can reproduce the parameters of a set of multigroup simulations up to 20% in radius and velocity. Observations of 15 SNe II are well fit by models with breakout radii <1014 cm. Eighteen SNe are typically more luminous, with observations at t ≥ 1 day that are better fit by a model with a large >1014 cm breakout radius. However, these fits predict an early rise during the first day that is too slow. We suggest that these large-breakout events are explosions of stars with an inflated envelope or with confined circumstellar material (CSM). Using the X-ray data, we derive constraints on the extended (∼1015 cm) CSM density independent of spectral modeling and find that most SN II progenitors lose Ṁ<10−4M⊙yr−1 up to a few years before explosion. We show that the overall observed breakout radius distribution is skewed to higher radii due to a luminosity bias. We argue that the 66−22+11% of red supergiants (RSGs) explode as SNe II with breakout radii consistent with the observed distribution of RSGs, with a tail extending to large radii, likely due to the presence of CSM.