Assuming you are Dutch or a permanent resident, having a gap of a few days between jobs has no real effect. You might miss a few days of income, and thus pension buildup, but that is it. Having no job is not relevant for your health insurance, as health insurance is personal and not tied to an employer(*). Some employer may provide a 'collectiviteitskorting' (collectivity discount) with a specific insurer, but for that only your employment status at or around the 1st of January is relevant (assuming the insurer even checks the collectivity).
As far as I know, this is not the case if you are not a permanent resident, but instead have a work permit/visa. However, I don't know the specifics for that situation, but I believe a gap might be problematic, check the conditions of your permit/visa and check with an immigration specialist.
However instead of having a real gap, consider using vacation days, if you have vacations days left at your old employer. It is usually advisable to use those. For that you have two options:
- You 'consume' the remaining vacations days at your old employer
- You ask your old employer to pay out the vacations days. Your new employer is legally required to allow you to take unpaid vacation days for the same number of days (you need proof of this from your old employer, and you need to use them within 6 months).
Generally taking option 1 is better than option 2, especially if you want a break between jobs.
*) Some employers may pay for health insurance, but, as far as I know, in that case end of your employment contract will just mean that you need to pay the insurance fee for the remainder of the year yourself.