I think the form answered your question. “… (if your state (not New Mexico) requires signature)”.
Philmont requires and relies on you having a completed and appropriately signed BSA Health and Medical form.
On your first day at Philmont during check-in you will have a medical re-check; height, weight, blood pressure, Health and Medical form review and you must show all prescription and OTC medications you will be carrying / taking while at Philmont. All prescription medication must be in their prescription bottle and listed on your Health and Medical form with appropriate signatures.
I hope this helps.
“Philmont should be enjoyed, not endured.” Moonshine