Consider two other translations:
Young's Literal Translation tries to copy the original Greek as closely as possible:
Only worthily of the good news of the Christ conduct ye yourselves, that, whether having come and seen you, whether being absent I may hear of the things concerning you, that ye stand fast in one spirit, with one soul, striving together for the faith of the good news, (YLT)
The literal Greek uses "one spirit" and "one soul" to refer to the life and body of God's church, acting in harmony, as a single entity.
The New Living Translation tries to convey the meaning in modern English:
Above all, you must live as citizens of heaven, conducting yourselves in a manner worthy of the Good News about Christ.
Then, whether I come and see you again or only hear about you, I will know that you are standing together with one spirit and one purpose, fighting together for the faith, which is the Good News. (NLT)
In today's language, "one spirit and purpose" conveys the same idea.
But note that in neither case does "spirit" have any supernatural connotations.
It has the same sense as when people (even atheists) tell others to "get into the spirit of the game" when they see them not really trying or enjoying it.
As for Seventh Day Adventists, Ellen G. White herself used this verse to convey an important message about the importance of unity among the faithful:
Ellen White's Shortest Testimony Arrives
Ten minutes before the meeting was to open, a Western Union messenger came to the door and inquired: “Is Mr. Campbell here?”
Campbell said, “Yes,” and reached out for the telegram addressed to him. Opening it he found this message: “Philippians 1:27, 28. (Signed) Ellen G. White.”
It was a testimony, her shortest testimony ever. Opening their Bibles to the reference given, they read:
“Only let your conversation be as it becometh the gospel of Christ: that whether I come and see you, or else be absent, I may hear of your affairs, that ye stand fast in one spirit, with one mind striving together for the faith of the gospel; and in nothing terrified by your adversaries: which is to them an evident token of perdition, but to you of salvation, and that of God.”
Elder Campbell reported:
That settled the question. That was a communication from Sister White that we needed right at that moment. God knew we were holding that meeting, and that we had a group of scared men, and that we needed help from Him, and so He gave us the message that came straight to us in the nick of time. It sounded pretty good to us.— Ibid., 14, 15.
In spite of the fact that every conceivable step was taken by the opposition to block the work of reorganization, the meeting was conducted successfully and adjourned somewhat after 11:00 P.M. Elder Campbell read again at that meeting Ellen White's counsel to safeguard the Tabernacle. It carried “great weight with the congregation” (G. W. Amadon to WCW, March 15, 1907).
The next morning George Amadon wrote to W. C. White at Elmshaven:
With much joy I hastily pen you a few lines. Many thanks for the telegram. How appropriate was the scripture. Well, the church held the adjourned meeting last evening. It was half-past eleven before we got home. There was a persistent and unreasonable opposition to every step taken.— Ibid.
Amadon reported that three fourths of the congregation voted for the articles and bylaws.
The Lord through His servant had sent warning messages. Faithful men heeding these messages had moved forward dramatically and with faith. The Battle Creek Tabernacle was saved for Seventh-day Adventists.
— Ellen G. White: The Later Elmshaven Years: 1905-1915 (vol. 6) — Ellen G. White Writings