Are any bonds broken when NO2 becomes N2O4?
No bonds are broken, but there are several isomers of N2O4.  The unpaired electron can be on either the N or the O, for each NO2 molecule. The long bond between the monomers involves these two unpaired electrons.  

See sections 7-1 to 7-3 of Bonding in Electron-Rich Molecules: Qualitative Valence-Bond Approach via Increased-valence Structures for more information.

According to First-Principles Study of the Role of Interconversion Between NO2, N2O4, cis-ONO-NO2, and trans-ONO-NO2 in Chemical Processes J. Am. Chem. Soc., 2012, 134 (31), pp 12970–12978:  

there is no barrier to the bonding of two monomers to form the planar isomer with the N-N bond.