There are two aspects to your question: can you to use your uni email and should you use your uni email.
The first is relatively simple to check with the HR, or the equivalent/corresponding, office of the university. At my university the rules is that you are not explicitly prohibited from using the uni mail for personal matters, however since it's a state university and university mail is technically an official public document, anyone (random average Joe) can actually demand to see your entire mail record (It should be added that the person needs to motivate it, but for example a journalist it's not a problem at all).
In other words, you are not explicitly prohibited to use your mail for personal reasons but you cannot keep it personal.
The second aspect is whether you should use that address. That's more difficult to answer as there are several different variables there. If you apply for a job in the industry it might be a good idea not to use an email that will expire, like a university mail. The reasoning is that if they contact you two years later for your dream job, the mail will be lost in the ether abyss and your misfortune will be someone else's lucky day. But I can understand the reasoning behind your motivation as well, it might be useful to show that you are already within the system.
either way, good luck!