German health insurance set-up for freelancers, shareholders of GmbH, UG

It is mandatory to have a health insurance in Germany while you are staying there.
You have two options:
private health insurance, or
public health insurance.
Private health insurance
If you are a young and healthy person with no “negative” health records, you have a good chance to get a private health insurance for about 200 - 250 EUR per month (like Hanse Merkur or Allianz) depending on your age, marital status (with children or without), etc. The rates get usually higher and higher over the years. For older persons it is not easy to change from a private health insurance into a public one.
Public health insurance
The public health insurance starts with the minimal monthly rate. They assume that you are earning, at least, a nominal income of about 2.178,75 EUR (2016) per month, so the calculation of the minimal monthly rate is based on this assumption. It amounts to about 398 EUR (day care insurance included), depending on your choice of a public health insurance company (Technische Krankenkasse, AOK, DAK, etc.). Some of them charge you with additional fees. Most of them have higher fees.

Every year you must send your income tax notice for the past year to the public health insurance company to enable them to calculate and recalculate your rates, for the future and for the past. If it takes the tax office too long to issue your income tax notice for the past year, the public health insurance company will ask you to give them an estimation of your expected income for the current year. Based on your statement about your (expected) annual income, they will fix a basic rate to be paid by you every month.
It is important that you should not overestimate your expected annual income because, in the case that you earn less than you expect, you are not going to be paid back any difference in your health insurance rates.
So if you state an expected annual income of 26.145,00 EUR (2016) for the current year, you will have to pay 14,6 % of 2.178,75 EUR = 318,10 EUR + 51,20 EUR day care insurance = roughly 369,30 EUR health insurance rate per month. But most of the insurance companies have higher rates because of the above mentioned additional fees.
If at the end of the year you should have earned more than you had stated at the beginning of the year, you will have to pay a difference to a higher health insurance rate based on your higher annual income. 
For example: if you should have earned, say, 50.850,00 EUR / 12 months = 4.237,50 EUR per month, you will have to pay the difference to the higher rate to the health insurance.
Here is the example:
4.237,50 * 15,9 % = 618,68 EUR + 99,58 EUR day care insurance = 718,26 EUR per month
Difference to be paid to the public health insurance company: 
718,26  – 369,30 = 348,96 * 12 months = 4.187,52 EUR
It is safer to start the year with a smaller annual income estimate, and a smaller health insurance rate, then keep money on a saving account if your income is growing, because if you start with a higher insurance rate, but the year turns out to be rather a “poor” one, you will not get back the “overplus” from the health insurance company.
The top basis for the calculation of public health insurance rates is an annual income of about 50.850,00 EUR (2016) , which changes a bit every year and depends on where you live in Germany. If you should have this or a higher annual income, you will have to pay the highest rate, but the rates will not increase further with your income.
Health insurance and income tax
In your income tax return, you can deduct the cost of the health insurance (personal deductions) from your taxable income. 
Attention: the health insurance rates can be deducted from your taxable income, not directly from the income tax.