The author has a curious interpretation of this verse, which quite frankly I have not heard before. Rather than interpreting this verse as referring to the lineage and lifespan of Melchizedek, the author applies this verse to the Levitical priesthood. It is best to quote the author at this point:

[H]e, Melchizedek, was without father, without mother, in the order of the Levitical priesthood. Without descent from the loins of Levi: Having neither beginning of days nor end of life; that is, he neither began nor ended his life or office in the regular line of the priesthood; but, in all these respects being made like unto the Son of God, whose descent was from Judah not from Levi, abideth a priest continually, . . . having neither successor nor predecessor in the sacred office. (98)

If the author of Hebrews was really intending to refer to the Levitical priesthood in this passage, this seems to me to be very odd language to use. Rather, I think, the author is cleverly using midrashic exegesis to argue that, since there is no mention of Melchizedek's lineage, birth, or death, he must not have had any lineage, nor did he experience birth or death. Hence, Melchizedek's priesthood is perpetual, since he never experienced death (what became of Melchizedek the author never ventures to surmise).

Now, the fact that Jesus did have a lineage from Judah (7:14) could be problematic for the analogy to be applied to Jesus. But I think the author of Hebrews must have Jesus' non-earthly or eternal existence in view, since he later says that Jesus became a priest according to the requirement of an "indestructable life" (7:16). Jesus "abides forever" (7:24) and "always lives" to make intercession (7:25). Moreover, passages such as 1:8, 1:12 and 13:8, which suggest the eternality of Jesus, must also be kept in view.