IANAL, but from my understanding, a EULA is a Contract of Adhesion https://www.law.cornell.edu/wex/adhesion_contract_contract_of_adhesion
And the clickwrap EULAs and TOSs you click through everywhere are reasonably safe for you to use: https://ilt.eff.org/index.php/Contracts:_Click_Wrap_Licenses
But, if you are recording user "click through" information in your own database, there are a few more aspects to your EULA that concern possible protections and liabilities to you.
By gathering personal information, you can be more certain the EULA has been clicked (or even read, god-forbid) by someone a bit more real than your average software clicker-and-user. The amount of info you gather depends on your needs, and could also include an email response from that user to confirm details and the EULA. That would seem to add more protection to you from the EULA.
And, when you say "your system," it sounds like they are logging in and accepting the EULA on a server you control. If so, you will have IP access logs for users, in addition to their EULA.
But, gathering that information could be a liability. A web user who clicks on a EULA sets a cookie in their browser or in a preference file on their hard-drive. You are gathering personal information and storing it yourself or at a third party, and you need to store it securely if it is anything more than name, email and phone. And ditto your own server; if the records are on the same server as the software and the user logins, you should be sure they are separate and secure.
And, of course, it is illegal to store many types of financial information without complying with strict state and federal laws. But if you are selling the software, you could also have the EULA integrated into a sales receipt and/or have a condition of them checking off they have accepted the EULA before they get a receipt.
Some/all of those options would make it hard for a customer to prove they didn't know what they were getting and dissuade them from complaining when faced with different types of proof showing they knew about the EULA.