As none of the chemists have picked up on this one, I will venture an answer.
First, I am answering on the assumption that this question relates to proteins as it was originally posted on Biology SE and you have previously asked chemical questions about proteins.
Second, I would have thought that you would be aware that weak interactions in proteins are usually categorized in three different types. The section on non-covalent bonds in Berg et al. states:
The three fundamental noncovalent bonds are electrostatic interactions, hydrogen bonds, and van der Waals interactions. They differ in geometry, strength, and specificity. Furthermore, these bonds are greatly affected in different ways by the presence of water.
So I find it strange that you wish to make an electrostatic interaction (aka an ionic bond) involving integral positive and negative charges into a hydrogen bond which can be regarded as a sort of poor man’s ionic bond as it involves only partial positive and negative charges. To quote Berg et al. again:
The relatively electronegative atom to which the hydrogen atom is covalently bonded pulls electron density away from the hydrogen atom so that it develops a partial positive charge (δ+). Thus, it can interact with an atom having a partial negative charge (δ-) through an electrostatic interaction.
The glutamic acid and arginine side-chains will be ionized at physiological pH because of the values of their ionization constants — the position of the equilibrium between the ionized and unionized forms. One extra hydrogen ion in the sea of hydrogen ions in the aqueous environment is not going to affect this one iota.
You should also be aware that the aqueous environment (in this case the surface of a globular soluble protein) is most unfavourable to the formation of hydrogen bonds between residues because of the competition of polar water molecules. To quote once more:
A hydrogen atom of water can replace the amide hydrogen atom as a hydrogen-bond donor, whereas the oxygen atom of water can replace the carbonyl oxygen atom as a hydrogen-bond acceptor. Hence, a strong hydrogen bond between a CO group and an NH group forms only if water is excluded.
You should read the whole of this section (which also applies to ionic bonds) to appreciate how these bonds are most important in the hydrophobic interior of proteins.
In summary, the answer to your question is “No hydrogen bonding will occur — if anything there will be ionic bonding, but in an aqueous environment this will be disrupted.”
You clearly have a reason for wanting glu and arg to form hydrogen bonds. Perhaps you could explain why.