2 bits of advice. Don't ask me any questions about my car. I know it has 4 wheels, and runs on gas. More than that, I don't know (I lied, I know it has a cassette deck), and I don't care. Never take financial advice from a car salesman. Why would you?
The lease is a red herring.
You have identified a car that you are willing to buy for $25,000. You have a 60 month plan to finance it. The more you put down, the lower the payments will be, and the interest rate is almost certain to be high enough that I'd agree with Pete should be avoided, paid down as quickly as possible. If you put $0 down, you will have 0% equity, more or less. $25K down, 100% equity. It's nonsense and a smokescreen in between two gaslights to suggest otherwise. Especially since the goal is to have no debt in 5 years.
What he told you makes zero sense, and you can tell him I said so.
Your salesman is either purposely lying, or is extremely confused. Both parts of the claim are wrong:
...that much down is not a good idea...
Actually, paying the entire amount down would be better to avoid interest. The only time you really ought to purposely put less down than you can afford is if the interest rate is 0%, or if you need cash-flow for an investment with a better return.
I will have no equity at the end of 5 years.
By definition, with a 5 year loan, after 5 years, you'll own the car. So unless the car is worth $0 after 5 years, you will definitely have some equity.
The only thing I can think of is maybe the salesman was confusing leasing with buying. If you were leasing again, then you would have no equity at the end of the lease, and putting a lot of money down is generally a bad idea with a lease.
Final Tip: make sure you shop around to see if you can get a similar car to yours for a better price than your buyout. The only benefits of buying your car is you typically avoid the return fee (approx $400-500) and if you have any expensive damage you won't have to pay to have it repaired.
There are many variables in your question. Assuming that the $25k cost is equivalent to or less than the residual value, then buying it outright might be a good deal. According to Carfax a new vehicle value drops more than 20% in the first year then loses roughly 10% of value annually for the next four years, which means that the car would be worth as little as 40% of original purchase price. How does the $25k cost compare to that depreciation of your car?
As for putting an additional $7k towards the $25k, that would reduce your monthly payment and overall debt which could reduce your APR. For that $25k you can probably buy yourself a much nicer car coming off a lease, like a BMW or Audi, than to refinance your car.