If you don't mind a bit of HTML copy/paste...
Settings > Templates > Packing Slips
Under Actions, click the 3 dots. You will need to work on a copy and give it a new name. That way if anything goes wrong, you can't lose your default. Also, this allows you to choose a different template when you go to print them based on whether you need a gift note or not.
This is the code we use. Your usage may vary, depending on which field your gift notes push to. Ours go to "Notes from Buyer." Under Order Footer, I have:
<table cellspacing=0 cellpadding="2" border="0" style="width:100%" class="footer">
<td rowspan="4" class="notes">
<div>[Notes from Buyer]</div>
By the way, under Order Header, Order Items Header, and Order Items you can also delete anything you don't want. For example, for our template that includes Notes from Buyer, we have deleted the codes that allow any prices to show on the packing slip. PLUS, you can download a sample before saving any changes.