How do the rapture and the millennium fit within Foursquare doctrine?

Good question! 

Foursquare doctrine is generally premillennial as it was formulated in a time when the Dispensational Premillennial position was the dominant eschatology (end-times doctrine) of Bible-believing Christians. Yet, the focus of Foursquare doctrine is not on the specific timing of the rapture, nor on the essential nature of the millennium (areas of significant diversity of opinion). 

The following is an excerpt from the official Foursquare Doctrinal statement. Note what it emphasizes and what it doesn't! 

  "We believe that the second coming of Christ is personal and imminent; that He will descend from Heaven 1in the clouds of glory with the voice of the archangel and with the trump of God; and that at this hour, which no man knoweth beforehand, the dead in Christ shall rise, then the 2redeemed that are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air, and that so shall they ever be with the Lord; that also seeing that a thousand years is as a day with the Lord, and 3that no man knoweth the hour of His appearance, which we believe to be near at hand, each day should be lived as though He were 4expected to appear at even, yet that in obedience to His explicit command, 5“Occupy till I come,” the work of spreading the gospel, the sending forth of missionaries, and the general duties for the upbuilding of the church 6should be carried on as diligently, and thoroughly, as though neither ours nor the next generation should live in the flesh to see the glorious day. 

Scripture References: 1. (1 Thessalonians 4:16,17); 2.(Titus 2:12,13); 3. (Matthew 24:36,42,44); 4. (Hebrews 9:28); 5. (Luke 19:13); 6. (Luke 12:35-37)" 

We can all agree that Jesus is coming back (suddenly, at a time that we do not know) and that we need to be doing His work until he returns. All other issues surrounding the timing of Christ's return are secondary and should not be dogmatically held grounds for divisiveness. 

The dominant theme of even the Book of Revelation is not a timetable for the return of Christ, but a spiritual encouragement to the Christian believer to "overcome" the idolatry and immorality so rampant in the world, rather than allowing themselves to be overcome by an anti-Christ world-view. Certainly an important issue today!