My contract ended March 29th but I have yet to sign anything to renew my contract nor have I received any letter of extension from the management. I took a leave on the 30th and did not show up on the 31st since I was not feeling well. I was told to come back to work on the 3rd of April with a Medical Certificate which I think may not be needed anymore since I have decided not to renew my contract for this company at all because of personal reasons. I have failed to give notice before my contract ended since I was also expecting to renew my contract but because of family and career matters, I have only decided not to renew on the 31st of March. Should I write a letter to the management? Is a 1-month rendered work still applicable if I am not resigning, since I have already ended my contract on March 29th? I am new to this sort of thing so I need your advice. Thank you. I appreciate it.
You generally need to comply both with what your contract says and with local laws. I suggest reading up on both carefully. We can't help you there since we don't know your contract or your location.
Is it alright to show up April 3 and tell employers I am not going to sign?
In most cases this is fine. If they never presented you with a contract, they never asked you "would you like to keep working here?". No contract = no work. You may not even legally be allowed to work (for example workplace insurance may not cover you anymore).
I was told to come back to work on the 3rd of April with a Medical Certificate
This you should do unless you have a strong reason not to. It's likely required to close out the paperwork for March
Which country? Laws, and customs, may differ.
In the US, I'd say that if your contract wasn't renewed, your last day is the day that the contract runs out. No notice period is needed because that was set up front. If either side wanted to continue the relationship, extension should probably have been discussed before the contract lapsed. But it may have slipped peoples' attention. They could still re-hire you on a new contract, so you should still ask, assuming you want to continue working with them.
Depending on the contract, there may be deliverables you still need to give them, despite the contract ending. Read your agreement. You may also want to talk to them about a short extension to cover transferring any knowledge they still need.
Part of being a successful consultant, I've been told, is always having a plan for what your next engagement might be when this one ends. Another part is realizing that you need to charge enough not only to cover the fact that you aren't getting a company benefits package, but to cover reasonable periods between contracts.
If your work contract ended on March 29th than starting March 30th you are unemployed. You don't need to do anything to severe your ties to your company, the relationship has ended already.
If both sides are willing it might be possible to start a new contract that backdates the employment start to March 30th but that requires both sides to be interested.