Changing jobs in blue card

Germany has compulsory health insurance and your current insurance cannot end unless you have a new one (or leave the country permanently).
If your employer stops paying half of the fees (the other half is automatically deducted from your salary), you will have to pay both halves from your own pocket. (Since the fee depends on your salary, it will be lower if you have no income in that period).
If you are eligible for unemployment benefits, the fee is automatically deducted from that. If not, your health insurer will contact you about the payment. If you don't want to wait, you can contact your insurer already.

Thanks a lot for sharing the information.
I will have to check with my current company as how i can take the contract .
Although i am planning to keep a minimum duration between two jobs mostly Sat and Sunday only so will check if the contract can end on Sunday and from Monday anyways my new contract will start

Varun, I think there is a misconception on your side: Health insurance in Germany is never a "contract" arranged or owned by the employer - it is your membership (that you must have signed up and have the paperwork for), which is not connected to your employment status.
Unless you are in a strange and rare non-standard constellation, you do not need to do anything. But if you want to, you can contact your health insurer. Your employer cannot help with this (although they must know which is your insurer, since they transfer your monthly salary deductions and their own contributions to them)!

The current insurer is a Private insurance company and the new insurer which I am applying for is TK once my new contract will start. I will check with my current private insurance as how I would need to pay for the days post my current contract end date or of my company will pay for the same and can be deducted from my fnf settlement

Changing from a private insurer to a public one (TK) is not easy.
As I said, the matter is very complicated, but in short:
Above a threshold of approx. €5400/month salary, you can choose between public and private insurers - but once in the private scheme, it is not normally possible to return to public.
Below that threshold, you must be publicly insured (no choice).
So, what you describe is only possible if you earned above the threshold in your old job and below in the new. If that is not the case, you need another option - e.g. just stay with the current private insurer.

To confirm what Beppi wrote; insurance is not made through an employer; they just have an obligation to pay half of the premium. Thus changing a job doesn't mean necessitate changing insurers – and once on a private plan it is not usually allowed to change to a public one. The insurance should just continue like before but the employer paid half of the premium gets taken over by the new employer.

And if there is somehow a short time between  the 2 jobs then it just means that one pays all of the premium for that short period of time; no big deal.  But this doesn't mean a lapse in coverage. But if one does manage to change insurance like mentioned; then one only needs to see that the coverage starts from the date the other ends. But again, the new employer might not be obliged to calculate their contribution until employment begins; but this is independent of the insurance coverage.

Anyway, this subject has been extensively covered before in multiple threads - and the poster being in Germany already should have easy access to such information.