It sounds like the company should be treating this as a wage.
Your accountant is thinking of having you taxed as an S-Corp to save on self-employment tax, which amounts to 15.3% of your income, but requires you to pay yourself through payroll and start a business taxed as an S-Corporation. The savings can be substantial, but make sure you're paying yourself a reasonable salary and that you're prepared to file corporate and payroll tax returns.
There is the self-employed health insurance deduction, or you can deduct on Schedule A as an employee, health insurance not run through an employer plan already etc. The former is typically the larger deduction given a 7.5% AGI floor for the Schedule A medical expenses to overcome, unless your employer has a plan, in which case you are likely getting the full deduction also.
Charitable contributions would show up on Schedule A as itemized deductions as well. In the absence of enough itemized deductions, you would simply take the standard deduction anyways (currently just under $6,000). The tax treatment of your income typically has no bearing on your ability to deduct charitable contributions, so long as you claim the income (meaning W2, 1099, K-1 ... it doesn't matter for charitable donations which you get).
You may be able to find other business expenses to deduct. Here's a good place to start looking (IRS Publication 535, Business Expenses):
(Mileage, unreimbursed corporate expenses you paid, computer equipment you had to buy for yourself, etc.)