"Free MgO" is just the amount of MgO added to the green body refractory during the batching process. This is typically done to form MgAl2O4 spinel in-situ. There is no "test" for this, as the batching process is typically known or reported.
If this is an unknown refractory (say from a competitor), it depends on if the condition of the refractory. Is it used, fired, a green body etc.? Also, it is difficult due to the addition of preformed spinel added to refractories. Does your refectory contain any preformed spinel?
Depending on the state of the refractory, you can do X-ray diffraction (XRD) to quantify the amount of MgO and MgAl2O4 by Rietveld analysis. Alternatively, EDS can give the total amount of Mg and just assume it is an oxide. Both of these techniques do not give you the "free MgO" content due to the fact that you need to know the state of the refractory as well as if there is any preformed spinel. If it is a green body, XRD and Rietveld can give you free MgO due to no reactions occurring during heating, regardless of the potential preformed spinel additions. If this is not the case, you can only make an inference, which I can elaborate on more if you'd like.
Actually, we used preformed fused spinel in fabricating refractory materials. My query is "how can we locate free MgO in the preformed fused spinel if the total conversion is not possible." What's the method to quantify free MgO in sample.
You should be able to do this fairly easily with XRD and performing a Rietveld analysis (or similar analysis). This should give you the weight percentage of each phase, and you will have to assume that all the detected MgO is free MgO.
One caveat to this is sample representation in the measured sample. You will have to make sure that the sample is homogeneous.
Good luck! And please let me know if you have questions on the analysis.