Green card holders,
It would seem to me that you have two issues; the residency requirements for retaining your legal immigrant status and/or gaining citizenship, and the residency requirements for taxation. Fortunately your goal is simple in both cases: you want to show the world that you reside in the U.S. permanently. That means get a driver's license, a social security number, buy a house or sign a year-long lease, put your kids in school, file U.S. tax returns as a resident, open bank accounts, etc. etc.
Nowhere on form I-90 (green card renewal) does it ask how many days you have been out of the country. It does say in the information material that if you ever leave the country for a continuous period of one year or more without filing N-470, you will be considered to have abandoned your legal immigrant status. So, you shouldn't have any problem renewing your green card as long as you don't do that. When applying for naturalization (citizenship), then you do have to show that you have been in the country for at least one-half the time (at least eighteen months of the last three years in the case of spouses). As long as you are not in a big hurry to become a citizen, though, you can just live happily as a green card holder until you satisfy the above requirement.
In both cases it is important that you do not become a legal resident of another country. This means that you pay taxes in the U.S. as someone who lives in the U.S. ;i.e. do not exclude from your taxable income the first $85,000 on form 2555 like non-residents can do. If the country in which you are working has a double-taxation agreement (most countries do), then most likely you will be able to not pay income tax in that country. In any case, you can file the Foreign Tax Credit form 1116 with your U.S. taxes in order to get a credit for foreign taxes that you do have to pay.
I'm not a lawyer or anything like that, but I have worked in Europe and lived in the U.S. for a few years.