Statements I: Only a few Goggles are Glares. All Specs are Glasses. Every Specs are Goggles means All Specs are Goggles.

Several Goggles are Specs: Every Specs are Goggles means All Specs are Goggles. Thus Some Goggles are Specs. So, conclusion I is true.
Some Glares being Glasses is a possibility: Only a few Goggles are Glares means Some Goggles are Glares and Some Goggles are not Glares. All Specs are Glasses. Every Specs are Goggles means All Specs are Goggles. The Goggles which are Glares can be Glasses. So, conclusion II is true.
Both conclusions I and II follow. Thus, statement I is correct.
Statement II: A Little Glares are Glasses means Some Glares are Glasses. No Glasses is Goggles. Only a few Glares are Specs.

Several Goggles are Specs: As there is no direct relation between Goggles and Specs. So, conclusion I is false.
Some Glares being Glasses is a possibility: A little Glares are Glasses means Some Glares are Glasses. So, conclusion II is false.
Neither conclusion I nor II follows. Thus, statement II is incorrect.
Statement III: No Glares is Goggles. All Specs are Glasses. Very few Glares are Glasses means Some Glares are Glasses.

Several Goggles are Specs: As there is no direct relation between Goggles and Specs. So, conclusion I is false.
Some Glares being Glasses is a possibility: Very few Glares are Glasses means Some Glares are Glasses. So, conclusion II is false.
Neither conclusion I nor II follows. Thus, statement III is incorrect.
Statement IV: Some Goggles are Glares. Only a few Specs are Goggles. Handful of Specs are Glasses means Some Specs are Glasses.

Several Goggles are Specs: Only a few Specs are Goggles means Some Specs are Goggles and Some Specs are not Goggles. Thus, Several Goggles are Specs. So, conclusion I is true.
Some Glares being Glasses is a possibility: As there is no direct relation between Glares and Glasses. There is a possibility of Some Glares being Glasses. So, conclusion II is true.
Both conclusions I and II follow. Thus, statement IV is correct.