An extensive body of research related to immigrants in a variety of countries has documented a "healthy immigrant effect" (HIE). When immigrants arrive in the host country they are healthier than comparable native populations, but their health status may deteriorate with additional years in the country. HIE is explained through the positive self-selection of the health of immigrants and the positive selection, screening and discrimination applied by the host countries. In this paper we study the health assimilation of immigrants within the context of selection and migration policies. Using SHARE data we are able to compare Israel and Europe that have fundamentally different migration policies. Israel has virtually unrestricted open gates for Jewish people around the world, who in turn have ideological rather than economic considerations to move. European countries have selective policies with regards to the health, education and wealth of migrants, who self-select themselves. Our hypothesis is that the HIE, evidenced in many countries will not be found in Israel. Instead, immigrants to Israel may arrive with lower health than that of natives and improve their health with residence in the country, due to the universal health coverage and generous socio-economic support of the government. Our results provide evidence that a) immigrants to Israel have compromised health and suffer from many health ailments upon arrival, making them less healthy than comparable natives. Their health does not improve for up to twenty years of living in Israel, after which they become similar to natives; b) immigrants to Europe have better health than natives upon arrival and up to eleven years since arrival in the host country, after which they are not significantly different than natives. Our results are important for policy.