The tax issue is whether there is a basis in the money. Basis essentially means, was there tax already paid on the money, or was it tax-free. If you pay medical expenses with HSA, FSA, or insurance money, that is un-taxed money (insurance money was never taxed, while HSA and FSA money was deducted from your taxable income when you made the contribution) and you can't deduct a tax you never paid. Gifted money was already taxed, because even though you did not pay tax on the income, the giver did, and the giver can't take a tax deduction for a personal gift, even if the recipient is needy.
Money is also "fungible". This means that once you receive the cash and put it in your bank account, it becomes mingled with all your other money and loses its "identity". You could just as easily say that you are using the gifts to pay your rent or mortgage and your own money to pay the medical bills.
Because money is fungible, and because the giver already paid tax on their income before they used some of it for the gift, you are allowed to deduct the medical expenses (subject to the other usual rules) even though you are paying with gifted money.
