I am proud to be American. I have reason to be. My many family branches can all be traced back to before the Revolution. Although we were very poor until recently, we fought against England in the Revolution and fought in the Civil War, among other things.
However, I know many people I would count at least as patriotic or more patriotic than I am. They wave the American flag even more than I do. Almost exclusively, they come from families that were legal immigrants who came during the ninth century. Whatever nation they came from and even though they have still have their own unique culture that is part of the wider American culture, they wave the American flag as enthusiastically as I do.
What is being part of your country? I do not know the answer to this question specifically, but I would suggest it involves being proud of the values your country (or your culture) tries to represent, being proud of the many good deeds it has done in the past, and being proud of the many great figures it has had. In America specifically, we most visible patriots tout our Declaration of Independence, our Constitution, the many great figures that founded America (George Washington, Benjamin Franklin, Thomas Jefferson, and John Adams, to name a few) and the great figures that came after (like Abraham Lincoln). Who cares if the patriot Paul Revere's swift ride overnight to warn the rebel militia in Lexington and Concord of British troops nearby became a more inflated legend? Paul Revere did do this as a patriot and his legend remains part of the American story. We won the battle that came after at least partially because of his warning.
When immigrants came to our country in the nineteenth century, we consciously tried to make them American--not just refugees fleeing their own country. That's one of the reasons, I'd argue, that our country is so rich in its culture. We all celebrate Saint Patrick's day today. Budweiser, for all their quality (it started in 1876 and the story is that Budweiser made good beer until we amended our Constitution to ban alcohol for a decade), is Germanic in origin. Bagels were invented in Poland and came from the Polish Jewish community. All are part of the wider American culture now. And each of these cultures that spread to America still have their own culture today. Many people are as proud of their roots as they are being American. It's not a mutually exclusive thing.
Today this is no longer the case. We do not try to make a conscious effort to "Americanise" the immigrants who come to America. This has led to much political divisiveness for a very good reason. If you immigrate to America, you must accept being American or that sets you in conflict with joining the wider American community. It's quite simple--if you don't accept being American, you set yourself in conflict with Americans who have been here generations. We have to tie ourselves in knots to accept your culture that you uniformly push on us AND reject our own culture. America welcomes you, aye, but that should require you to welcome becoming American.
Europe has the same problem now that the Syrian refugee crisis has come upon them. In a conversation I had last night, one of my friends here cited a poll that more British Muslims were proud to be British than your normal Brit. I cannot verify the poll, but I'm not surprised. Each of their families made the CONSCIOUS decision to become part of Great Britain and the British culture. They learned English. They accepted the queen or king that was ruling at the time and the cultural significance of that. They accepted the idea of Parliament. Aye, they may have changed the culture a bit, but they did not change the wider British culture. All those Brits who are running off to Syria are the sons and daughters of Muslim families who moved here a generation or more ago, not those immigrants who immigrated. And radical clerics are a new thing, not an old thing. Muslims do not serve alcohol in their restaurants because of their religion, but if you want a pint you can easily get one elsewhere.
My point here is that one of the worries many European nations have about the Syrian refugee crisis is that Western culture might fall because of it. They do have an argument. That's one of our troubles with illegal immigration. However, this problem can be solved. It's quite simple, really: do what we did in the nineteenth century. Do what those British Muslims did generations ago. Don't reject. However, acclimate them as well. Germany, stress Bismark, Frederic the Great, and how you became Germany. France, you have so many figures: Charles de Gaul, Napoleon, and Charlemagne to name a few. Your culture is significant. Invite these people to become part of your country and your culture instead of force them to stay part of an entirely different culture independent of your country. Don't keep them refugees. Make them part of a country they can be proud of living in.
Then, I can assure you, in a few generations these people--patriots now--will be waving your flag as enthusiastically as you do.