Here's my personal opinion, having only briefly skimmed MUFHH, and that, many years ago ...
I don't necessarily have a problem with cautiously reading Protestant academics; biblical journals, commentaries, etc. in today's modern academic world are usually pretty denomination-neutral, so you won't find overtly "Protestant" ideas for the most part. I was just telling someone recently that one of the best commentaries on Hebrews that I've ever read was written by an Anglican scholar, who wrote 600 pages on the subject of the Eucharist as the centerpiece of Hebrews, and showed how the early liturgies of the Church support this. Absolutely fantastic work.
But MUFHH is a devotional work; and this is an entirely different arena. This is spirituality. This is not a realm, IMO, where we should be entrusting ourselves to Protestant ideas.
If you're looking for a good devotional, read St. Francis DeSales' Introduction to the Devout Life, or read the book that St. Francis DeSales himself carried around, Spiritual Combat.
After you've thoroughly mastered those books (good luck!) and have begun to live out the principles explained therein in your daily life (this will take you the rest of your life, BTW), then you can look at something like MUFHH - and when you do, you will be so utterly bored with Oswald Chambers' deficient outlook on spirituality that you probably won't bother.
If you're looking for something a bit lighter than St. Francis DeSales, then try a book like Day by Day with the Early Church Fathers (see here); I think a Protestant compiled and published this, but at least it's Catholic in its content.
EDITED TO ADD: You might also consider going through St. Alphonsus Ligouri's The Glories of Mary; short, bite-size chapters, with prayers at the end of each section.