Can =NH2+ in Arginine's side chain form hydrogen bonding at physiological pH?
Yes, the arginine side chain is an excellent hydrogen bond donor. Charged N-H groups are even better hydrogen bond donors than the corresponding neutral N-H groups. Here is an article that gives examples of arginine side chains in hydrogen bonds. It showcases bidendate hydrogen bonds (i.e. two hydrogen bonds from the arginine side chain) with carboxylates and with nitrogenous bases of nucleic acids.

Specifically for proteins, if the hydrogen bond donor is positively charged (e.g. Arg+) and the hydrogen bond acceptor is negatively charged (e.g. Glu-), you have a combination of ionic and hydrogen bond interaction, which is particularly strong. It is often referred to as a salt bridge, such as in this example. If you take a close look at the arginine side chain, you will see that it is protonated (i.e. positively charged) and interacts with two negatively charged carboxylate groups.