Question 783031
Expanding and simplifying...
(a+1)(b+1)(c+1)=
(a+1)(bc+b+c+1)=
abc+ab+ac+bc+a+b+c+1=
ab+bc+ac+a+b+c+2

Since their product is 1, the only possibilities are a,b,c={-1,1}  (why?)
Furthermore, there must be an odd number of positives. To find the minimum we can ignore the case a=b=c=1 - that would be the maximum.

Case1
if say a=b=1, c=-1, then

ab+bc+ac+a+b+c+2=
1-1-1+1+1-1+2=
2

Case2
if say c=1, a=b=-1, then

ab+bc+ac+a+b+c+2=
1-1-1-1-1+1+2=
0

Minimum value is 0