Can I deduct my non-dependent son's hospital bills for 2024? I paid these bills and can I deduct them on my taxes?

No, unfortunately you cannot deduct the medical expenses of a non-dependent. See: Whose Medical Expenses Can You Include?

However, if he was unemployed, were you supporting him?  Perhaps he was your dependent.  Review the rules for an other dependent here:  Rules for Claiming Dependents on Taxes, but note that the $5,050 maximum income is waived for purposes of determining if he was a dependent for medical purposes.

Yes, you can enter the expenses you paid for him with you out of pocket funds.  Enter as an itemized deduction medical expense.

Health care insurance premiums, including dental and vision insurance premiums, and other medical expenses that you paid with out of pocket funds and were not reimbursed are an eligible medical expense that you can deduct using Schedule A for itemized deductions. However, only your total medical expenses that are greater than 7.5% of your Adjusted Gross Income (AGI) can be deducted. Your total itemized deductions reported on Form 1040 Schedule A must be greater than the standard deduction for your filing status to have any tax benefit.

Standard deductions for 2024

Single - $14,600 add $1,950 if age 65 or older
Married Filing Separately - $14,600 add $1,550 if age 65 or older
Married Filing Jointly - $29,200 add $1,550 for each spouse age 65 or older
Head of Household - $21,900 add $1,950 if age 65 or older

To enter your medical expenses -

Click on Federal Taxes (Personal using Home and Business)
Click on Deductions and Credits
Click on I'll choose what I work on (if shown)
Scroll down to Medical
On Medical Expenses, click the start or update button

Or enter medical expenses in the Search box located in the upper right of the program screen. Click on Jump to medical expenses