In regards to posture, in Principles of the Flute, Hotteterre said: “As it is necessary to add grace to skill as far as possible in order to attain perfection in the practice in which you want to succeed, I will begin this treatise with an explanation of the posture you must assume when playing the flute” (Hotteterre, 1).
Correct posture for playing the Baroque flute is the same as it should be when you’re playing the modern flute. You must stand (or sit) straight and tall with the head raised up slightly (never tilted down or too far up, as this negatively affects intonation and tone clarity), and elbows and shoulders not raised. Your left wrist must be bend inwards and your left arm held close to your torso. When standing, the left foot needs to be forward with the weight of the body on the right hip—this helps reduce tension. Try to avoid unnecessary movement, which can impair your sound and intonation. Remember that with the Baroque flute, every little movement or adjustment will cause a reaction in your sound quality (be it better or worse) and especially your intonation.
As far as hand position is concerned, both Hotteterre and Quantz stressed that the flute must be held naturally and comfortably, without undue tension. The left wrist must be bent slightly (to allow the third finger to reach more easily), with the left thumb directly under the first finger, and all fingers curved slightly, with the fleshy pads of the fingers being what is covering the holes.
The right wrist must also be a bit bent, with the right thumb bent and curved slightly outwards with the tip beneath the first finger of the right hand. The fingers, again, must be slightly curved, but not so much that the tips of the fingers are being used to cover the holes—you always want the pads of the fingers to do that. When the key is not in use, the little finger can be placed on the flute between the 6th hole and the foot joint.
Hotteterre emphasised how difficult it is to properly explain hand position in writing (which is why I strongly believe that this website shouldn’t replace a tutor, just supplement one), so in his treatise, he often referred the reader to a drawing of himself holding the flute. I’ve included that drawing for your reference.
The embouchure you use for the Baroque flute requires even flexibility than the modern flute (this is because the tuning on a Baroque flute is not as stable, and thus requires movement on almost every note). It must be very relaxed, with the bottom lip spread, and the top lip tense enough to control the air stream, but not so tense that it thins the tone. To start, the lower rim of the mouth hole should be near the middle of the red portion of the lower lip, with the top lip out slightly further, as to direct the air halfway into the flute, and halfway over the top edge of the mouth hole. From D’’ down to D’ (and, generally, as you crescendo), the lips will be drawn back a bit, and the opening will slightly increase in length and width, and the bottom lip will roll out to cover more of the mouth hole. From D’’ upwards (and, generally, as you decrescendo), the lower lip will push out slightly more than the upper, and the embouchure will become a bit smaller and more narrow. Never press the lips together too tightly, as this will thin the tone.
Of course, everyone’s lips, teeth, and instruments are different, so you might find that you need to alter your embouchure slightly to produce the desired sound. However, the general mechanics of forming an embouchure and producing sound will stand.