Settling the debt will not negatively affect your credit. Having the collection on your credit was the negative effect. If the collection is still reporting as opened and updating on your report every month, it's best to take care of it because it will hold down your scores until it's addressed.
Settling the debt will get the creditor to report the account as closed, and you will stop action on the creditor's part in trying to get money from you. The balance owed should update to $0 with the remark "paid in full for less than balanced owed" OR "settled in full".
Asking the creditor to delete the collection usually requires a full payment of the debt. Deletion from your credit will have it show that this collection nor the account ever existed, which is why creditors typically won't settle with you for something like this.
When did the Verizon collection happen? Depending on how old it is, you may be able to call the credit bureaus and request an early exclusion.
Early exclusion would not work for you then if it happened in October 2014.
If it's with another collection agency there's only a matter of time before they start reporting. If that is the case, I would pay in full and ask Jefferson Capital not to report. Your HBC consultant is right, if they report, it would drop his score because it would present itself as a new collection.
It would also be helpful if you type Jefferson Capital in the search boards of the forums. You can see if others had issues with them and how they handled it.