Non-citizens should be allowed to serve on juries and participate in most civic activities, even be granted limited voting rights, as some states have done. This is a good way of making an open and welcoming environment for would-be citizens, and just because one is not a citizen does not mean they will not be a civically responsible person.
No, non-citizens should not be allowed to serve on juries, because non-citizens do not have the same knowledge of, and respect for, American culture. A non-citizen has no responsibility or moral responsibility to uphold the constitution. They may have a different experience that leads them to have a different opinion about our laws. Citizen-jurors are better.