Let φ : R n × [ 0 , ∞ ) → [ 0 , ∞ ) \varphi : \mathbb R^n\times [0,\infty )\to [0,\infty ) be such that φ ( x , ⋅ ) \varphi (x,\cdot ) is an Orlicz function and φ ( ⋅ , t ) \varphi (\cdot ,t) is a Muckenhoupt A ∞ ( R n ) A_\infty (\mathbb R^n) weight uniformly in t t . In this article, for any α ∈ ( 0 , 1 ] \alpha \in (0,1] and s ∈ Z + s\in \mathbb {Z}_+ , the authors establish the s s -order intrinsic square function characterizations of H φ ( R n ) H^{\varphi }(\mathbb R^n) in terms of the intrinsic Lusin area function S α , s S_{\alpha ,s} , the intrinsic g g -function g α , s g_{\alpha ,s} and the intrinsic g λ ∗ g_{\lambda }^* -function g λ , α , s ∗ g^\ast _{\lambda , \alpha ,s} with the best known range λ ∈ ( 2 + 2 ( α + s ) / n , ∞ ) \lambda \in (2+2(\alpha +s)/n,\infty ) , which are defined via L i p α ( R n ) \mathrm {Lip}_\alpha ({\mathbb R}^n) functions supporting in the unit ball. A φ \varphi -Carleson measure characterization of the Musielak-Orlicz Campanato space L φ , 1 , s ( R n ) {\mathcal L}_{\varphi ,1,s}({\mathbb R}^n) is also established via the intrinsic function. To obtain these characterizations, the authors first show that these s s -order intrinsic square functions are pointwise comparable with those similar-looking s s -order intrinsic square functions defined via L i p α ( R n ) \mathrm {Lip}_\alpha ({\mathbb R}^n) functions without compact supports, which when s = 0 s=0 was obtained by M. Wilson. All these characterizations of H φ ( R n ) H^{\varphi }(\mathbb R^n) , even when s = 0 s=0 , \[ φ ( x , t ) := w ( x ) t p   for\ all   t ∈ [ 0 , ∞ )   and   x ∈ R n \varphi (x,t):=w(x)t^p\ \textrm {for\ all}\ t\in [0,\infty )\ \textrm {and}\ x\in {\mathbb R}^n \] with p ∈ ( n / ( n + α ) , 1 ] p\in (n/(n+\alpha ), 1] and w ∈ A p ( 1 + α / n ) ( R n ) w\in A_{p(1+\alpha /n)}(\mathbb R^n) , also essentially improve the known results.