Some variant of "if all the illegal aliens in the U.S. were white Europeans illegal immigration wouldn't be an issue" or variants is one of the lesser-used, more ludicrous, more divisive talking points from those who support massive immigration. If they actually thought it through those supporters probably wouldn't use it. Here are some things to consider:
* The claim is racially divisive, falsely pretending that all or almost all opposition to illegal immigration is racially-based.
* If all the illegal aliens were white Europeans, illegal immigration supporters couldn't play the race card. The fact that corruption is at the heart of their support would be much more evident to more people.
* If all the illegal aliens were white Europeans, would the left-leaning establishment rise to defend them as they do now with largely-Hispanic illegal aliens? Would, for instance, the mainstream media constantly promote anti-American bills like the DREAM Act if the main beneficiaries were from Sweden (and, as in the current case, those mainly harmed are minorities)? Would the mainstream media offer PIIPPs in support of students from Denmark?
* Would that same left-leaning establishment support racial power groups representing illegal aliens if those illegal aliens were white? Would the "National Council of White Europeans" be as favored by the establishment as the corresponding National Council of La Raza?
* In a Zogby poll of Mexicans in Mexico, fully 58% of Mexicans said that the U.S. Southwest rightfully belongs to Mexico. It's not difficult at all to find such a sentiment in the U.S., nor is it difficult to find a similar sentiment that those of Indian origin have a right to move anywhere within the Americas. The percentage of Europeans who think they have a race-based claim on U.S. territory is probably at or very close to zero (there might be some obscure theories I'm not aware of).
* However, some countries might think they do have some legal but not race-based claim to our land. For instance, Russian politician Vladimir Zhirinovsky has hinted at re-claiming Alaska for Russia (New York Times, December 19, 1993, link). If millions of Russians moved to Alaska, most Americans would realize the danger of it falling into their hands and would - free of the race card - be able to oppose it.
* If we built a land bridge to Norway, exactly how many Norwegians would try to cross it? Probably not many; in fact, they would probably need to worry about an influx of Americans going to take advantage of their more generous welfare state. With less successful European countries (like, for instance, Greece), there would probably be a flow into the U.S., but most would probably decide to stay in Greece out of national pride (obviously, since we don't have such a land bridge it's hard to tell).
* If all the illegal aliens were from one European country - say Germany or France - and there were an assimilation problem as there is with those from Latin America, would Americans tire of hearing "pour le service en francais, appuyer deux" or similar? Would Americans look at the issues in Canada with Quebec for guidance and decide that a large ethnic minority that has assimilation problems isn't worth it? And, wouldn't they - free of having the race card played - be able to engage in attrition in order to reduce the problem?
* Would a European country that had a land bridge to the U.S. take advantage of it to get rid of disfavored minorities, such as Turkish guest workers or the Roma? Wouldn't that be somewhat similar to how most illegal aliens from Mexico aren't from the top levels of their racial caste system? Would "liberals" support that too?