Can nitrogens in urea be protonated?
Reagents/solutions that are acidic enough will protonate the urea group, just not where you think. The most basic site is the oxygen. Perhaps this is a surprise. 

The lone pairs on the nitrogen atoms are delocalized (see resonance depiction below).
 

The oxygen atom has the most electron density. Below is an electron density map, where ${\mathrm{\color{red}{red}}}$ represents areas of high electron density and $\mathrm{\color{blue}{blue}}$ represents areas of low electron density. You can see the area around the oxygen atom is red and yellow, while the areas around the nitrogen atoms are green.



This image is from "The effect of intermolecular hydrogen bonding on the planarity of amides" by James A. Platts, et al. published in Phys. Chem. Chem. Phys., volume 2012, issue 14, page number 11944-11952, DOI:10.1039/C2CP41716B. The image is copyright the Royal Society of Chemistry and released under a Creative Commons Attribution 3.0 Unported license.

Protonating the oxygen atom leads to a cation where the positive charge is delocalized (better, more stable).



Protonating one of the nitrogen atoms leads to a cation where the positive charge is localized (worse, less stable).