The Shulhan Aruch (Orah Haim 440; listen to audio recording for precise citation) writes that if a gentile enters a Jew’s home during Pesah with Hametz, the Jew is not required to force the gentile to leave. He adds that even if the Jew sees the gentile’s Hametz, the gentile may remain in the home with the Hametz, and no violation is entailed. The Mishna Berura (commentary by Rav Yisrael Meir Kagan of Radin, 1839-1933), in Se’if Katan 16 (listen to audio recording for precise citation), explains that the Torah prohibition of "Lo Yera’eh" forbids seeing one’s own Hametz during Pesah, but not seeing the Hametz belonging to somebody else. And, the Sages were not concerned in such a case that the Jew may mistakenly eat the Hametz, since the Hametz is in the gentile’s possession, and the Jew would certainly not try to steal the Hametz from the gentile. Therefore, if a non-Jew visits a Jew’s home on Pesah and has Hametz with him, the Jew may allow him to remain. However, the Mishna Berura adds that after the gentile leaves, the Jew must ensure that the gentile takes all his Hametz with him. If, for example, the gentile had come in with a bag of pretzels, the Jews should make sure that he takes all the pretzels with him when he leaves.
The Shulhan Aruch notes in this context that although a Jew may allow a non-Jew with Hametz in his home during Pesah, he may not eat together with a non-Jew at the same table if the non-Jew is eating Hametz. Even if they make a "Hefsek" ("separation") by eating on separate tablecloths or placemats, the Jew may not eat together with a non-Jew who eats Hametz. When it comes to a Jew who eats meat and another who eats dairy foods, Halacha permits them to eat together with separate mats. When it comes to eating on Pesah with a gentile who eats Hametz, however, this is forbidden. The Mishna Berura explains this Halacha based on the principle that Hametz is forbidden even "Be’mashehu," meaning, in the smallest quantity. When two people eat together, even using separate mats or cloths, it is nearly impossible to ensure that no crumbs from one person’s food end up on the other’s plate. Therefore, it is forbidden on Pesah to eat together with a gentile who eats Hametz, even using separate mats.
However, the Mishna Berura adds, if a gentile ate Hametz at a table, the Jew may eat at that table after the gentile finishes eating, provided, of course, that he cleans the table very thoroughly before he eats to ensure there are no crumbs or residue left over from the Hametz. Thus, for example, if a coworker ate his sandwich on a table in the office, the Jew may then eat his "kosher for Passover" lunch after thoroughly cleaning the table.
Summary: It is permitted to have a gentile enter one’s home with Hametz during Pesah, but he must ensure that the gentile brings all his Hametz with him when he leaves. It is forbidden on Pesah to eat together with a gentile who eats Hametz, but one may eat on the table where a gentile ate Hametz after he finished eating, assuming he cleans it very well to ensure that no particles of Hametz remain.
